Theorems · Theorem · field theory

IntermediateField.adjoin_subset_adjoin_iff

∀ (F : Type u_1) [inst : Field F] {E : Type u_2} [inst_1 : Field E] [inst_2 : Algebra F E] {F' : Type u_3}
  [inst_3 : Field F'] [inst_4 : Algebra F' E] {S S' : Set E},
  ↑(IntermediateField.adjoin F S) ⊆ ↑(IntermediateField.adjoin F' S') ↔
    Set.range ⇑(algebraMap F E) ⊆ ↑(IntermediateField.adjoin F' S') ∧ S ⊆ ↑(IntermediateField.adjoin F' S')

Defined in: Mathlib.FieldTheory.IntermediateField.Adjoin.Defs
Cited by: 1 results in Mathlib
Foundations: Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field Field Algebra Field Algebra

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Algebrastatement and proof · cited by 11,388
RingHomstatement · cited by 10,189
SetLike.coestatement and proof · cited by 8,199
Fieldstatement and proof · cited by 7,404
Algebra.algebraMapstatement and proof · cited by 4,706
Set.rangestatement and proof · cited by 4,705
LE.le.transproof · cited by 3,151
IntermediateFieldstatement · cited by 988
IntermediateField.adjoinstatement and proof · cited by 382
Set.union_subsetproof · cited by 71

Cited by1

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IntermediateField.adjoin_adjoin_leftproof · cited by 6